Existence of Laplace Transforms Before continuing our use of Laplace transforms for solving DEs ...

41 Pages·2015·2.55 MB·English

Laplace Transform Theory - 1
Existence of Laplace Transforms
Before continuing our use of Laplace transforms for solving DEs, it
is worth digressing through a quick investigation of which functions
actually have a Laplace transform.
A function f is piecewise continuous on an interval t ∈ [a, b] if
the interval can be partitioned by a finite number of points a = t0 <
t1 < · · · < tn = b such that
• f is continuous on each open subinterval (t , t ).
i−1 i
• f approaches a finite limit as the endpoints of each subinterval
are approached from within the subinterval.
In other words, f is continuous on [a, b] except for a finite number of
jump discontinuities. A function is piecewise continuous on [0,∞) if
f(t) is piecewise continuous on [0,N] for all N > 0.
Laplace Transform Theory - 2
Problem. Draw examples of functions which are continuous and
piecewise continuous, or which have different kinds of discontinuities.
One of the requirements for a function having a Laplace transform is
that it be piecewise continuous. Classify the graphs above based on
this criteria.
Laplace Transform Theory - 3
Another requirement of the Laplace transform is that the integral
∫
∞
−st
e f(t) dt converges for at least some values of s. To help
0
determine this, we introduce a generally useful idea for comparing
functions, “Big-O notation”.
Big-O notation
( )
at
We write f(t) = O e as t → ∞ and say f is of exponential
order a (as t → ∞) if there exists a positive real number M and a
at
real number t0 such that |f(t)| ≤ Me for all t > t0.
Laplace Transform Theory - 4
|f(t)|
Lemma. Assume lim exists. Then
at
e
t→∞
|f(t)|
lim < ∞
at
e
t→∞
( )
at
if and only if f(t) = O e as t → ∞. □
Problem. Show that bounded functions and polynomials are of ex-
ponential order a for all a > 0.
Laplace Transform Theory - 5
2
t
Problem. Show that e does not have exponential order.
Problem. Are all the functions we have seen so far in our DE solu-
tions of exponential order?
Laplace Transform Theory - 6
The final reveal: what kinds of functions have Laplace transforms?
Proposition. If f is
• piecewise continuous on [0,∞) and
• of exponential order a,
then the Laplace transform L{f(t)}(s) exists for s > a.
The proof is based the comparison test for improper integrals.
Laplace Transform of Piecewise Functions - 1
Laplace Transform of Piecewise Functions
In our earlier DE solution techniques, we could not directly solve
non-homogeneous DEs that involved piecewise functions. Laplace
transforms will give us a method for handling piecewise functions.
Laplace Transform of Piecewise Functions - 2
Problem. Use the definition to determine the Laplace transform of

2 0 < t ≤ 5 ,

f(t) = 0 5 < t ≤ 10 ,

 4t
e 10 < t .
Laplace Transform of Piecewise Functions - 3

2 0 < t ≤ 5 ,

f(t) = 0 5 < t ≤ 10 ,

 4t
e 10 < t .
The Unit Step Function - 1
We would like avoid having to use the Laplace definition integral if
there is an easier alternative. A new notation tool will help to simplify
the transform process.
The Heaviside step function or unit step function is defined
{
0 for t < 0,
by u(t) :=
1 for t ≥ 0.
Problem. Sketch the graph of u(t).
t